It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing Δ₁¹-comprehension (...) schema would already be inconsistent. In the present paper, we show that this is not the case. (shrink)

Leo Harrington showed that the second-order theory of arithmetic WKL 0 is ${\Pi^1_1}$ -conservative over the theory RCA 0. Harrington’s proof is model-theoretic, making use of a forcing argument. A purely proof-theoretic proof, avoiding forcing, has been eluding the efforts of researchers. In this short paper, we present a proof of Harrington’s result using a cut-elimination argument.

Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this (...) extended system. (shrink)

Levinas's ethics of other-centered service has been criticized at the theoretical level for failing to offer a conception of moral agency adequate to ground its imperative and at the practical level for encouraging self-hatred. Levinas's explicit resistance to the incorporation of the phrase "as yourself" in the Judaeo-Christian love command might seem to validate the critics' complaints. The author argues, on the contrary, that Levinas does offer a strong and compelling conception of moral agency and that his ethics, properly understood, (...) does not entail self-abnegation. Levinas's attempt to counter excessive and manipulative self-concern and self-inflation by insisting on the dependent and situational position of the self has been wrongly overinterpreted as an abandonment of the self and its just claims. The author seeks to establish a more balanced understanding by focusing attention on the "ethics of welcome," on Levinas's distinctive conception of passivity, and on the role of "the third" in all human relations. (shrink)

Kierkegaard's "Works of Love" provocatively presses for a reconsideration of impartiality, partiality, and equality. Past readings of this text have typically (1) criticized its focus on the abstract category of "human being," ignoring its attention to distinctiveness and difference; (2) defended it from the charge of abstraction by accenting its treatment of distinctiveness and difference, playing down its assumptions about the "essentially" human; (3) acknowledged its emphases on both essence and difference, arguing that they are incompatible and irreconcilable; or (4) (...) acknowledged both emphases, assuming they are compatible without exploring or accounting for the apparent incompatibility. As a means of resolving this seeming inconsistency, I will focus on Kierkegaard's recommendation of moral blindness and its implications for moral vision, and I will argue that "Works of Love" contains resources for an understanding of impartiality that allows moral attention to concrete difference. (shrink)

Frege's "Grundgesetze der Arithmetik" is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege's Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the "Grundgesetze" is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this (...) extended system. (shrink)

After identifying contrasting formulations of the practical postulates of reason in Kant's second critique, I analyse the context of each formulation, showing both how the postulate of the of God is consistent with Kant's understanding of a significant transition arising from practical needs as well as how the postulate of the existence of God can be seen as a acting out a . My goal is to re-examine Kant's view of the relation between the practical and theoretical employments of reason (...) in order to distinguish clearly between what Kant sees as required of the moral believer as opposed to permitted. (shrink)

Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is an impassable barrier in the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson's theory of arithmetic Q. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted in Q but (...) also what cannot be so interpreted. We explain some of these results and rely on them to discuss Nelson's position. (shrink)

What has become known as the ‘faith/history’ problem for historical religions like Christianity centres on the attempt to combine the ontological decisiveness, for faith, of an historical event characterized as an actual Incarnation of God with the epistemological indifference, or irrelevance, of historical information about that event which is decisive for faith. Without the former there is nothing to be related to or personally appropriated; without the latter faith is rendered vulnerable to the vagaries of historical research. Soren Kierkegaard's Climacus (...) writings – the Philosophical Fragments and the Concluding Unscientific Postscript – provide a classic formulation of the inherent tension. Twentieth-century variations on the same theme are found in the writings of Paul Tillich and Rudolf Bultmann, among others. In what follows I want to highlight an element in Kierkegaard's Climacus account which crucially distinguishes his attempt to deal with the faith/history relation from other attempts to deal with it, and thus offers what I suggest is a novel approach to overcoming the tension. This element – effectively a kind of a priori or conceptual proof – has been neglected, and whatever one's final judgement on its contribution to the problem as a whole, a consideration of it and its implications is, at the very least, integral to a full understanding of the thought-experiment in the Fragments . Moreover, in the process, useful light can be shed indirectly on twentieth-century attempts to deal with the faith/history problem. (shrink)

One of the most basic functions of human language is to convey who did what to whom. In the world's languages, the order of these three constituents (subject [S], verb [V], and object [O]) is uneven, with SOV and SVO being most common. Recent experiments using experimentally elicited pantomime provide a possible explanation of the prevalence of SOV, but extant explanations for the prevalence of SVO could benefit from further empirical support. Here, we test whether SVO might emerge because (a) (...) SOV is not well suited for describing reversible events (a woman pushing a boy) and (b) pressures to be efficient and mention subjects before objects conspire to rule out many other alternatives. We tested this by asking participants to describe reversible and non-reversible events in pantomime, and we instructed some participants to be consistent in the form of their gestures and to teach them to the experimenter. These manipulations led to the emergence of SVO in speakers of both English (SVO) and Turkish (SOV). (shrink)

Stemming from Uexkull’s fundamental concepts of Umwelt and Innenwelt as developed in the biosemiotic approach of Ferreira 2010, 2011, the present work models mathematically the semiosis of cognition and proposes an artificial cognitive architecture to be deployed in a robotic structure.

The dynamics inherent to the life activity of all living systems presents itself in the form of regular patterns viewed by the observer as taking place in an extended timeline. Routines, rituals and celebrations, each in their own way, are defined by the typical cyclical behavioural patterns exhibited by individuals embedded in specific semiospheres. The particular nature of these semiospheres will determine the distinct patterns of behaviour to be adopted in different life contexts so that existential functions are fulfilled. The (...) restricted and protected family circle normally provides the initial learning environment where the definition of the individual’s Umwelt (von Uexküll 1909, 1934; Ferreira, Biosemiotics 3(1):107–130, 2010, 2011) his/her meaningful world, starts to take shape. This learning process comprehends the progressive identification of distinct physical entities, the development of basic patterns of physical and social behaviour- the incorporation of basic routines and the assignment of meaning to particular events. (shrink)

Levelt, Roelofs & Meyer present a comprehensive and sophisticated theory of lexical access in production, but we question its reliance on binding-by-checking as opposed to binding-by-timing and we discuss how the timing of retrieval events is a major factor in both correct and errorful production.

Let be the fragment of elementary Peano arithmetic in which induction is restricted to -formulas. More than three decades ago, Parsons showed that the provably total functions of are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the -consequences of universal theories. We give a self-contained proof requiring only basic knowledge of mathematical logic.

An intermediate stage in Hrushovski’s construction of flat strongly minimal structures in a relational language L produces ω-stable structures of rank ω. We analyze the pregeometries given by forking on the regular type of rank ω in these structures. We show that varying L can affect the isomorphism type of the pregeometry, but not its finite subpregeometries. A sequel will compare these to the pregeometries of the strongly minimal structures.

In his major work on love, Works of Love, Kierkegaard clearly and robustly affirms the moral superiority of neighbourly love, and approves preferential love on one condition: that it serve as an instance of neighbourly love. But can an essentially preferential love be an instance of the essentially non-preferential neighbourly love? John Lippitt seems to think it can. In his paper “Kierkegaard and the problem of special relationships: Ferreira, Krishek, and the ‘God filter”’ he defends Kierkegaard’s position in Works of (...) Love against my criticism (as presented in my book Kierkegaard on Faith and Love); specifically, against my claim that in using Kierkegaard’s view of neighbourly love as a framework for understanding preferential love, one fails to account for the latter’s distinctive character. Lippitt claims that I misinterpret Kierkegaard’s position and, using what he calls ‘the God filter’, he attempts to show how adhering to Kierkegaard’s view of neighbourly love allows one to sustain the distinctiveness (and value) of preferential love. In what follows I will defend my interpretation of Kierkegaard’s position and explain why I take the view he presents in Works of Love to be problematic. Furthermore, in my aforementioned book I offer a Kierkegaardian model of love that does precisely what Lippitt seeks his ‘God filter’ model to do: namely, preserve the distinctiveness of preferential love while allowing its possible coexistence with neighbourly love. Thus, against the background of Lippitt’s criticism I will demonstrate this model again, in hope of clarifying the advantages this view offers. (shrink)